= Solution
The <flux average for a translating lubrication gap> follows by integrating the <pressure gradient> between the endpoints:
$$
0=6\int_{-1}^1\frac{\widehat h-2Q}{\widehat h^3}\,d\widehat x.
$$
If $2Q$ were smaller than the minimum height, the integrand would be strictly positive everywhere, contradicting zero total <pressure> change. If it were larger than the maximum, the integrand would be strictly negative. Therefore
$$
\boxed{\min\widehat h\leq2Q\leq\max\widehat h.}
$$
Equivalently,
$$
2Q=\frac{\int_{-1}^1\widehat h^{-2}\,d\widehat x}{\int_{-1}^1\widehat h^{-3}\,d\widehat x}
$$
is a weighted average of the height with strictly positive weight $\widehat h^{-3}$. This also identifies the constant <flux> uniquely.
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